THE MAN, NOT THE METHOD three unknowns, for example as in ax2 + 2bxy -f cyz = ?«, where a&c,m are any given integers and it is required to discuss all integer solutions x, y of the equation. The point to be noted here is that the problem is stated and is to be solved entirely in the domain of the rational integers, that is. in the realm of discrete number. To fit analysis, which is adapted to the investi- gation of continuous number, to such a discrete problem would seem to be an impossibility, yet this is what Hermite did. Starting with a discrete formulation, he applied analysis to the problem, and in the end came out with results in the discrete domain from which he had started. As analysis is far more highly developed than any of the discrete techniques invented for algebra and arithmetic, Hermite's advance was comparable to the introduction of modern machinery into a medieval handicraft. Hermite had at his disposal much more powerful machinery, both algebraic and analytic, than any available to Gauss when he wrote the Disquisitiones Arithmeticae. With Hermite's own great invention these more modern tools enabled him to attack problems which would have baffled Gauss in 1800. At one stride Hermite caught up with general problems of the type which Gauss and Eisenstem had discussed, and he at least began the arithmetical study of quadratic forms in any number of un- knowns. The general nature of the arithmetical theory of forms' can be seen from the statement of a special problem. Instead of the Gaussian equation ax* + 2te/ -j- ct/2 = m of degree ta> in two unknowns (x9 y), it is required to discuss the integer solutions of similar equations of degree n in 5 unknowns, where n, s are any integers, and the degree of each term on the left of the equation is n (not 2 as in Gauss' equation). After stating how he had seen after much thought that Jacobi's researches on the periodicity of uniform, functions depend upon deeper questions in the theory of quadratic forms, Hermite out- lines his programmes. 'But, having once arrived at this point of view, the problems - vast enough - which I had thought to propose to myself, seemed inconsiderable beside the great questions of the general theory of forms. In this boundless expanse of researches which 503